Interatomic surfaces

At the heart of the AIM theory is the definition of an atom as it exists in a molecule. An atom is defined as the union of a nucleus and the atomic basin that the nucleus dominates as an attractor of gradient paths. An atom in a molecule is thus a portion of space bounded by its interatomic surfaces but extending to infinity on its open side. As we have seen, it is convenient to take the 0.001 au envelope of constant density as a practical representation of the surface of the atom on its open or nonbonded side because this surface corresponds approximately to the surface defined by the van der Waals radius of a gas phase molecule. Figure 6.15 shows the sulfur atom in SC12. This atom is bounded by two interatomic surfaces (IAS) and the p = 0.001 au envelope. It is clear that atoms in molecules are not spherical. The well-known space-filling models are an approximation to the shape of an atom as defined by AIM. Unlike the space-filling models, however, the interatomic surfaces are generally not flat and the outer surface is not necessarily a part of a spherical surface. 

Figure 6.15 Three-dimensional representation of the sulfur atom in SC12. This atom is bounded by two interatomic surfaces (IAS) and one surface of constant electron density (p = 0.001 au). Topologically, an atom extends to infinity on its nonbonded side, but for practical reasons it is capped. Each interatomic surface contains a bond critical point (BCP).

Having defined an atom in a molecule, we can, at least in principle, determine any of the properties of an atom in a molecule. The simplest to illustrate is the atomic volume, which is simply the sum of all the volume elements that occupy all the space defined by the interatomic surfaces and the p = 0.001 au contour. More exactly, it is the integral of all the volume elements dr over the atomic basin. If we denote the atomic basin by Si, then the volume of the atom is given by. 

An important advantage of the finite atoms defined by AIM is that they do not overlap, which is not generally true for orbital-defined atoms. Each atom has a sharp and well-defined boundary inside the molecule, given by its interatomic surfaces. The atoms fit exactly into each other, leaving no gaps. In other words, the shape and the volume of the atoms are additive. This is true also for other physical properties of an atom, such as the electron population and the charge, as seen in Table 6.2 and as indeed has been shown to be true for all other properties. (Bader 1990, Popelier 1999). 

Figure 6.17 Contour map of p in the interatomic surface associated with the CC bond critical point in ethene. The plane of the plot is perpendicular to the molecular plane. The C and two H nuclei are projected onto the plane of the plot to indicate the orientation of the molecule. We see that electronic charge is preferentially accumulated in the direction perpendicular to the molecular plane, giving an elliptical shape to the electron density in this plane.

Figure 6.18 Contour maps of the ground state electronic charge distributions for the period 2 diatomic hydrides (including H2) showing the positions of the interatomic surfaces. The outer density contour in these plots is 0.001 au. (Reproduced with permission front Bader [1990].)...

Figure 2. Contour maps of the electron density of (a) SCI2 and (b) H2O. The density increases from the outermost 0.001 au isodensily contour in steps of 2 x 10", 4 x 10", and 8 x 10" au with n starting at 3 and increasing in steps of unity. The lines connecting the nuclei are the bond paths, and the lines delimiting each atom are the intersection of the respective interatomic surface with the plane of the drawing. The same values for the contours apply to subsequent contour plots in this paper.

So far we have considered the shape of the electron density of a limited inner region of each atom but not of the complete atom. How do we find the shape of the complete atom In other words, how do we find the interatomic surfaces that separate one atom from another and define the size and shape of each atom The atoms in molecules (AIM) theory developed by Bader and coworkers (4) provides a method for doing this. 

For a homonuclear diatomic molecule such as Cl2 the interatomic surface is clearly a plane passing through the midpoint between the two nuclei—in other words, the point of minimum density. The plane cuts the surface of the electron density relief map in a line that follows the two valleys leading up to the saddle at the midpoint of the ridge between the two peaks of density at the nuclei. This is a line of steepest ascent in the density on the two-dimensional contour map for the Cl2 molecule (Fig. 9). 

The AIM theory provides a clear and rigorous definition of an atom as it exists in a molecule. It is the atomic basin bounded by the interatomic surfaces. The interatomic... 

The concept of a bond has precise meaning only in terms of a given model or theory. In the Lewis model a bond is defined as a shared electron pair. In the valence bond model it is defined as a bonding orbital formed by the overlap of two atomic orbitals. In the AIM theory a bonding interaction is one in which the atoms are connected by a bond path and share an interatomic surface. 

Another common method of representing the electron density distribution is as a contour map, just as we can use a topographic contour map to represent the relief of a part of the earth s surface. Figure 7a shows a contour map of the electron density of the SCI2 molecule in the Oh (xy) plane. The lines in which the interatomic surfaces, that are discussed later, cut this plane are also shown. Figure 7b shows a corresponding map for the H20 molecule. 

Also indicated by arrows are the two trajectories that terminate at the BCP in this symmetry plane. They are members of the infinite set of such trajectories that define the interatomic surface of zero-flux in Vp between the boron and fluorine atoms. 

The density is a maximum in all directions perpendicular to the bond path at the position of a bond CP, and it thus serves as the terminus for an infinite set of trajectories, as illustrated by arrows for the pair of such trajectories that lie in the symmetry plane shown in Fig. 7.2. The set of trajectories that terminate at a bond-critical point define the interatomic surface that separates the basins of the neighboring atoms. Because the surface is defined by trajectories of Vp that terminate at a point, and because trajectories never cross, an interatomic surface is endowed with the property of zero-flux - a surface that is not crossed by any trajectories of Vp, a property made clear in Fig. 7.2. The final set of diagrams in Fig. 7.1 depict contour maps of the electron density overlaid with trajectories that define the interatomic surfaces and the bond paths to obtain a display of the atomic boundaries and the molecular structure. 

Atomic volumes play an important role in relating physicochemical properties to biological effects. Most atoms in molecules are not entirely bounded by interatomic surfaces and an atomic volume is defined as a measure of the space enclosed by the intersection of the atom s zero-flux surfaces with some outer envelope of the density. The envelope with a value of 0.001 au is generally chosen as this has been shown to yield molecular sizes in good agreement with experimentally assigned van der Waals radii [16, 17]. A related property is the van der Waals surface area, which QTAIM determines by integrating an atom s exposed contribution to a molecule s isovalued surface. 

Till now, the AIM in QCT comes from an entirely topological origin. The single most important step forward in the theory was the realization that a quantum mechanical AIM coincides exactly with this topological atom [53]. The definition of an interatomic surface is given by... 

This equation means that the normal to the surface S, n(r), is orthogonal to the gradient of the electron density. In other words, the surface is parallel to Vp, or rephrased again, the surface consists of gradient paths. The interatomic surface is a bundle of gradient paths that terminate at the bond critical point at the center of the surface. 

Figure 3.11 Contour lines (gray) of the electron density, the molecular graph (black), and interatomic surfaces (black) in the dimeric structure (BH3-NH3)2 obtained at the MP2/6-31G level. Here the bond critical points are marked as squares and the ring critical points as triangles. The labels of the nuclei located in the mirror plane (the plane of the paper) are solid, and those that do not lie in this plane are open. (Reproduced with permission from ref. 16.)...

Since the electron density is a continuous function across the interatomic surface, the two atoms that form the surface must have the same distribution of electrons over this face. The most stable structures will be those which require the least amount of redistribution of electron density when the free atoms come together, that is, they will be formed between atoms that have similar surface electron densities. This idea is related to the valence matching principle (Rule 4.2) which states that the most stable bonds are formed between ions that have similar bonding strengths. The bonding strength is thus related to the surface electron density of the ion. 

Interatomic surface Zero-flux surface S Internuclear surface through which the flux of Vp(r) is zero (see equation 6)... 

Analysis of the electron density distribution p (r) of numerous molecules has revealed that there exists a one-to-one relation between MED paths, saddle points p and interatomic surfaces on the one side and chemical bonds on the other27,81,82. However, low-density MED paths can also be found in the case of non-bonding interactions between two molecules in a van der Waals complex82. To distinguish covalent bonding fron non-bonded or van der Waals interactions, Cremer and Kraka have given two conditions for the existence of a covalent bond between two atoms A and B8. 

Fig. 1. Water charge density in the molecular plane, (a) the contour map. The outermost contour has the value 0.0067 eA-3. The density increases almost exponentially for inner contours. The bond paths, the interatomic surfaces and the bond critical points are also indicated. (b) The relief map where the atom-cores are seen as peaks (reproduced with permission from Bader [2]).

Atomic volume Atomic charge Atomic dipole moment Atomic energy Interatomic surface Covalent bond... 

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